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I want to show the following equalities for harmonic sum

$$\sum_{k=1}^{\infty}\frac{(-1)^k}{k^3}=\lim_{n\to\infty}\int_0^{2n}\frac{-3}{x^4}([x]-2[\frac{x}{2}])dx$$

Any idea?

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    $\begingroup$ do you know how to prove this identity? $\endgroup$
    – Mxifeng
    Commented Dec 8, 2017 at 16:59
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    $\begingroup$ This question is in the level of bachelor student and suitable for Math.StackExchange . But when I was bachelor student in 2007, I wrote a book about 400 problems in Mathematical Analysis I,II based on questions of Putnam competition and Mathematical monthly journal. see page 146 problem 38 here drive.google.com/file/d/0B-JKfzn6I35qd29adTFwTVZHMmM/… (you can use google translate ) $\endgroup$
    – user21574
    Commented Dec 9, 2017 at 4:28

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